The key for understanding the previous MSE decomposition is to realise that $\theta$ represents a point variable with a fixed value and $\hat{\theta}$ represents a vector containing samples from the estimator (random variable).

As clarification, this formula does not apply to the case where $\theta$ represents a series of values and $\hat{\theta}$ contains element-wise estimations of these points.

A simple numpy example that verifies the previous formula might be useful for understanding this:

import numpy as np

# theta is the target value (point)
theta = 10
# theta_hat contains estimations of theta (vector of 1000 samples)
theta_hat = np.random.normal(theta,0.2,1000)

mse = np.mean(np.square(theta-theta_hat))

var = np.var(theta_hat)
bias = np.mean(theta_hat) - theta

print("MSE result:", mse)
print("Decomposed result:", var + bias*bias)

Which in my case returns 0.042668116306306667 in both cases. So, the decomposition stands, at least, up to the 18th decimal place in numpy.

The key for understanding the previous MSE decomposition is to realise that $\theta$ represents a point variable with a fixed value and $\hat{\theta}$ represents a vector containing samples from the estimator (random variable).

As clarification, this formula does not apply to the case where $\theta$ represents a series of values and $\hat{\theta}$ contains element-wise estimations of these points.

A simple numpy example that verifies the previous formula might be useful for understanding this:

import numpy as np

# theta is the target value (point)
theta = 10
# theta_hat contains estimations of theta (vector of 1000 samples)
theta_hat = np.random.normal(theta,0.2,1000)

mse = np.mean(np.square(theta-theta_hat))

var = np.var(theta_hat)
bias = np.mean(theta_hat) - theta

print("MSE result:", mse)
print("Decomposed result:", var + bias*bias)

Which in my case returns 0.042668... in both cases.

The key for understanding the previous MSE decomposition is to realise that $\theta$ represents a point variable with a fixed value and $\hat{\theta}$ represents a vector containing samples from the estimator (random variable).

As clarification, this formula does not apply to the case where $\theta$ represents a series of values and $\hat{\theta}$ contains element-wise estimations of these points.

A simple numpy example that verifies the previous formula might be useful for understanding this:

import numpy as np

# Θ is the target value (point)
Θ = 10
# Θ_est contains estimations of Θ (vector of 1000 samples)
Θ_est = np.random.normal(Θ,0.2,1000)

mse = np.mean(np.square(Θ-Θ_hat))

bias = np.mean(Θ)-np.mean(Θ_hat)
var = np.var(Θ_hat)

print("MSE result:", mse)
print("Decomposed result:", var + bias*bias)

Which in my case returns 0.042668116306306667 in both cases. So, the decomposition stands, at least, up to the 18th decimal place in numpy.

The key for understanding the previous MSE decomposition is to realise that $\theta$ represents a point variable with a fixed value and $\hat{\theta}$ represents a vector containing samples from the estimator (random variable).

As clarification, this formula does not apply to the case where $\theta$ represents a series of values and $\hat{\theta}$ contains element-wise estimations of these points.

A simple numpy example that verifies the previous formula might be useful for understanding this:

import numpy as np

# theta is the target value (point)
theta = 10
# theta_hat contains estimations of theta (vector of 1000 samples)
theta_hat = np.random.normal(theta,0.2,1000)

mse = np.mean(np.square(theta-theta_hat))

var = np.var(theta_hat)
bias = np.mean(theta_hat) - theta

print("MSE result:", mse)
print("Decomposed result:", var + bias*bias)

Which in my case returns 0.042668116306306667 in both cases. So, the decomposition stands, at least, up to the 18th decimal place in numpy.